HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  lnopl Structured version   Visualization version   GIF version

Theorem lnopl 28901
Description: Basic linearity property of a linear Hilbert space operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
Assertion
Ref Expression
lnopl (((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))

Proof of Theorem lnopl
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ellnop 28845 . . . . . 6 (𝑇 ∈ LinOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
21simprbi 479 . . . . 5 (𝑇 ∈ LinOp → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
3 oveq1 6697 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥 · 𝑦) = (𝐴 · 𝑦))
43oveq1d 6705 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥 · 𝑦) + 𝑧) = ((𝐴 · 𝑦) + 𝑧))
54fveq2d 6233 . . . . . . 7 (𝑥 = 𝐴 → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = (𝑇‘((𝐴 · 𝑦) + 𝑧)))
6 oveq1 6697 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 · (𝑇𝑦)) = (𝐴 · (𝑇𝑦)))
76oveq1d 6705 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)) = ((𝐴 · (𝑇𝑦)) + (𝑇𝑧)))
85, 7eqeq12d 2666 . . . . . 6 (𝑥 = 𝐴 → ((𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)) ↔ (𝑇‘((𝐴 · 𝑦) + 𝑧)) = ((𝐴 · (𝑇𝑦)) + (𝑇𝑧))))
9 oveq2 6698 . . . . . . . . 9 (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
109oveq1d 6705 . . . . . . . 8 (𝑦 = 𝐵 → ((𝐴 · 𝑦) + 𝑧) = ((𝐴 · 𝐵) + 𝑧))
1110fveq2d 6233 . . . . . . 7 (𝑦 = 𝐵 → (𝑇‘((𝐴 · 𝑦) + 𝑧)) = (𝑇‘((𝐴 · 𝐵) + 𝑧)))
12 fveq2 6229 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
1312oveq2d 6706 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 · (𝑇𝑦)) = (𝐴 · (𝑇𝐵)))
1413oveq1d 6705 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴 · (𝑇𝑦)) + (𝑇𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝑧)))
1511, 14eqeq12d 2666 . . . . . 6 (𝑦 = 𝐵 → ((𝑇‘((𝐴 · 𝑦) + 𝑧)) = ((𝐴 · (𝑇𝑦)) + (𝑇𝑧)) ↔ (𝑇‘((𝐴 · 𝐵) + 𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝑧))))
16 oveq2 6698 . . . . . . . 8 (𝑧 = 𝐶 → ((𝐴 · 𝐵) + 𝑧) = ((𝐴 · 𝐵) + 𝐶))
1716fveq2d 6233 . . . . . . 7 (𝑧 = 𝐶 → (𝑇‘((𝐴 · 𝐵) + 𝑧)) = (𝑇‘((𝐴 · 𝐵) + 𝐶)))
18 fveq2 6229 . . . . . . . 8 (𝑧 = 𝐶 → (𝑇𝑧) = (𝑇𝐶))
1918oveq2d 6706 . . . . . . 7 (𝑧 = 𝐶 → ((𝐴 · (𝑇𝐵)) + (𝑇𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))
2017, 19eqeq12d 2666 . . . . . 6 (𝑧 = 𝐶 → ((𝑇‘((𝐴 · 𝐵) + 𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝑧)) ↔ (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶))))
218, 15, 20rspc3v 3356 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶))))
222, 21syl5 34 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇 ∈ LinOp → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶))))
23223expb 1285 . . 3 ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇 ∈ LinOp → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶))))
2423impcom 445 . 2 ((𝑇 ∈ LinOp ∧ (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ))) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))
2524anassrs 681 1 (((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wf 5922  cfv 5926  (class class class)co 6690  cc 9972  chil 27904   + cva 27905   · csm 27906  LinOpclo 27932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-hilex 27984
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-lnop 28828
This theorem is referenced by:  lnop0  28953  lnopmul  28954  lnopli  28955
  Copyright terms: Public domain W3C validator